The lifespans of bears in a particular zoo are normally distributed. The average bear lives $48$ years; the standard deviation is $7$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a bear living between $62$ and $69$ years.
$48$ $41$ $55$ $34$ $62$ $27$ $69$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $48$ years. We know the standard deviation is $7$ years, so one standard deviation below the mean is $41$ years and one standard deviation above the mean is $55$ years. Two standard deviations below the mean is $34$ years and two standard deviations above the mean is $62$ years. Three standard deviations below the mean is $27$ years and three standard deviations above the mean is $69$ years. We are interested in the probability of a bear living between $62$ and $69$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the bears will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the bears will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of bears between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular bear living between $62$ and $69$ years is $\color{orange}{2.35\%}$.